Help with angle stops under my sink

I need to replace some VERY old hardware going from my angle stops under the bathroom sink, to the faucet above. The angle stops are really old and aren't the standard 1/2" size, so I'm having trouble finding the right line to replace the old stuff.

Obviously, the easiest thing to do would be to put in new angle stops, but plumber friend has advised against that, since the copper piping is so old and could easily snap. (I'm renting anyway, and just don't need that headache...)

So I'm left with finding out what I'm supposed to do with this ancient, non-standard angle stop. I've discovered that the thread on the angle stop fit's a 1/2inch FLARE fitting exactly. The only trouble there is that the angle stop doesn't have the flare "nipple" (not sure what its called - the shape of the flare male end), so it doesn't seal.

The way I see it, I have two choices: 1. find some kind of rubber washer or gasket to put into the end of the flare line which would seal it up. 2. find a regular, non-flare line that will fit the old angle stop.

Your other option is call your landlord and have him fix the leaky plumbing. ​

Part of the deal with this place, is that I take care of all this kind of stuff.

I've never heard of old copper snapping? I'll wait for others to reply. ​

The copper pipe is about 60 years old... it's pretty corroded...

A flared shutoff with copper coming out of the wall?
Even I haven't seen that​

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I know. It's pretty ancient stuff.

Actually, it's not really 1/2" flare coming out of the wall. It's a regular IP style fitting. It just so happens that the thread exactly fits a 1/2" Flare line. I tried using a flare line (like for a gas dryer or something like that) but it doesn't seal because of the shape of the flare line fitting at the end.

I was also thinking that it I could find some sort of solid, one-piece adapter to fit onto the angel stop, I could go from there. Does anyone know where I could get an adapter that has one side 1/2" FEMALE flare?

as a total novice to in this sort of thing, part of the trouble I'm having is how these things are measure and labled. For example, a 1/2" flare fitting is actually 3/4" across the threading. 1/2" IP is about 7/8" across the threading.

I guess what I need is something that is 3/4" across the threading (I just measured it) that has a regular IP style female end...